Pavel Exner (Prague)
Abstract: The topic of this talk are quantum graphs with the vertex coupling which does not preserve the time-reversal invariance. As a case study the simplest example with the asymmetry being maximal at a fixed energy will be analyzed. In this situation the high-energy scattering depends crucially on the vertex parity; we will demonstrate implications of this fact for spectral and transport properties in several classes of graphs, both finite and infinite periodic ones. In particular, we prove the Band-Berkolaiko universality for Kagome lattices with this coupling. Furthermore, we discuss other timeasymmetric graphs and identify a class of such couplings which exhibits a nontrivial PT-symmetry despite being self-adjoint; we also illustrate the role of the Dirichlet component in the vertex coupling and discuss spectrum of the Cairo lattice. Finally, we show how a square lattice with such a coupling behaves in the presence of a magnetic field when the two timeasymmetry effects compete. The results come from a common work with Marzieh Baradaran, Jirí Lipovský, and Milos Tater.
